Understanding Practical Optimization in Machine Learning

Egor Shulgin

Modern machine learning relies on practical optimization methods that often differ from the algorithms studied in classical analyses: they train submodels, clip or normalize gradients, use layer-wise non-Euclidean directions, or approximate costly inner computations. This thesis analyzes the objective, assumptions, update, and computation defined by these procedures, rather than those of simplified proxies. For submodel training, we show that Independent Subnetwork Training can converge to a shifted fixed point because sketching before differentiation biases the update for the original objective. Model-Agnostic Sparsified Training instead incorporates the sketch distribution into its objective, which makes sketched gradients unbiased and admits convergence guarantees. For private optimization, we show that adaptive quantile clipping retains bias at any fixed quantile, while suitable quantile schedules recover near-optimal nonconvex rates under heavy-tailed noise. We then combine smoothed normalization with error compensation and obtain distributed private convergence guarantees under standard assumptions, without assuming bounded gradients. For optimizers based on linear…